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Shared Qs (029)


  1. Question

    An ellipse is a set of points such that the total distance from (1) one focus to (2) the edge to (3) the other focus is constant for any point on the edge. The two foci (\(F_1\) and \(F_2\)) are a distance \(2c\) apart, and the total distance from \(F_1\) to the edge to \(F_2\) is a distance \(2a\).

    Let’s consider the example below with foci at \((-6.3,0)\) and \((6.3,0)\) and a covertex at \((0, 6)\):

    plot of chunk unnamed-chunk-2

    Evaluate \(2a\) and \(2c\).

    Now, let’s pick another point on the ellipse, with (approximate) coordinates (4.7, 5.05):

    plot of chunk unnamed-chunk-3

    The distance \(s\) is from \(F_1\) to the point on the edge. The distance \(w\) is from \(F_2\) to the point on the edge. Evaluate \(s\), \(w\), and the total distance (\(s+w\)).

    Remember, the total distance when connecting

    1. The first focus \(F_1\)
    2. Any point on the edge
    3. The second focus \(F_2\)

    equals \(2a\). Based on this information, find the location of the right vertex at (\(x\), 0) in the diagram below.

    plot of chunk unnamed-chunk-4



    Solution


  2. Question

    An ellipse is a set of points such that the total distance from (1) one focus to (2) the edge to (3) the other focus is constant for any point on the edge. The two foci (\(F_1\) and \(F_2\)) are a distance \(2c\) apart, and the total distance from \(F_1\) to the edge to \(F_2\) is a distance \(2a\).

    Let’s consider the example below with foci at \((-4,0)\) and \((4,0)\) and a covertex at \((0, 4.2)\):

    plot of chunk unnamed-chunk-2

    Evaluate \(2a\) and \(2c\).

    Now, let’s pick another point on the ellipse, with (approximate) coordinates (4.21, 2.89):

    plot of chunk unnamed-chunk-3

    The distance \(s\) is from \(F_1\) to the point on the edge. The distance \(w\) is from \(F_2\) to the point on the edge. Evaluate \(s\), \(w\), and the total distance (\(s+w\)).

    Remember, the total distance when connecting

    1. The first focus \(F_1\)
    2. Any point on the edge
    3. The second focus \(F_2\)

    equals \(2a\). Based on this information, find the location of the right vertex at (\(x\), 0) in the diagram below.

    plot of chunk unnamed-chunk-4



    Solution


  3. Question

    An ellipse is a set of points such that the total distance from (1) one focus to (2) the edge to (3) the other focus is constant for any point on the edge. The two foci (\(F_1\) and \(F_2\)) are a distance \(2c\) apart, and the total distance from \(F_1\) to the edge to \(F_2\) is a distance \(2a\).

    Let’s consider the example below with foci at \((-3.6,0)\) and \((3.6,0)\) and a covertex at \((0, 7.7)\):

    plot of chunk unnamed-chunk-2

    Evaluate \(2a\) and \(2c\).

    Now, let’s pick another point on the ellipse, with (approximate) coordinates (7.31, 3.93):

    plot of chunk unnamed-chunk-3

    The distance \(s\) is from \(F_1\) to the point on the edge. The distance \(w\) is from \(F_2\) to the point on the edge. Evaluate \(s\), \(w\), and the total distance (\(s+w\)).

    Remember, the total distance when connecting

    1. The first focus \(F_1\)
    2. Any point on the edge
    3. The second focus \(F_2\)

    equals \(2a\). Based on this information, find the location of the right vertex at (\(x\), 0) in the diagram below.

    plot of chunk unnamed-chunk-4



    Solution


  4. Question

    An ellipse is a set of points such that the total distance from (1) one focus to (2) the edge to (3) the other focus is constant for any point on the edge. The two foci (\(F_1\) and \(F_2\)) are a distance \(2c\) apart, and the total distance from \(F_1\) to the edge to \(F_2\) is a distance \(2a\).

    Let’s consider the example below with foci at \((-1.1,0)\) and \((1.1,0)\) and a covertex at \((0, 6)\):

    plot of chunk unnamed-chunk-2

    Evaluate \(2a\) and \(2c\).

    Now, let’s pick another point on the ellipse, with (approximate) coordinates (5.53, 2.53):

    plot of chunk unnamed-chunk-3

    The distance \(s\) is from \(F_1\) to the point on the edge. The distance \(w\) is from \(F_2\) to the point on the edge. Evaluate \(s\), \(w\), and the total distance (\(s+w\)).

    Remember, the total distance when connecting

    1. The first focus \(F_1\)
    2. Any point on the edge
    3. The second focus \(F_2\)

    equals \(2a\). Based on this information, find the location of the right vertex at (\(x\), 0) in the diagram below.

    plot of chunk unnamed-chunk-4



    Solution


  5. Question

    An ellipse is a set of points such that the total distance from (1) one focus to (2) the edge to (3) the other focus is constant for any point on the edge. The two foci (\(F_1\) and \(F_2\)) are a distance \(2c\) apart, and the total distance from \(F_1\) to the edge to \(F_2\) is a distance \(2a\).

    Let’s consider the example below with foci at \((-4.2,0)\) and \((4.2,0)\) and a covertex at \((0, 4)\):

    plot of chunk unnamed-chunk-2

    Evaluate \(2a\) and \(2c\).

    Now, let’s pick another point on the ellipse, with (approximate) coordinates (3.17, 3.35):

    plot of chunk unnamed-chunk-3

    The distance \(s\) is from \(F_1\) to the point on the edge. The distance \(w\) is from \(F_2\) to the point on the edge. Evaluate \(s\), \(w\), and the total distance (\(s+w\)).

    Remember, the total distance when connecting

    1. The first focus \(F_1\)
    2. Any point on the edge
    3. The second focus \(F_2\)

    equals \(2a\). Based on this information, find the location of the right vertex at (\(x\), 0) in the diagram below.

    plot of chunk unnamed-chunk-4



    Solution


  6. Question

    An ellipse is a set of points such that the total distance from (1) one focus to (2) the edge to (3) the other focus is constant for any point on the edge. The two foci (\(F_1\) and \(F_2\)) are a distance \(2c\) apart, and the total distance from \(F_1\) to the edge to \(F_2\) is a distance \(2a\).

    Let’s consider the example below with foci at \((-2.8,0)\) and \((2.8,0)\) and a covertex at \((0, 4.5)\):

    plot of chunk unnamed-chunk-2

    Evaluate \(2a\) and \(2c\).

    Now, let’s pick another point on the ellipse, with (approximate) coordinates (3.6, 3.3):

    plot of chunk unnamed-chunk-3

    The distance \(s\) is from \(F_1\) to the point on the edge. The distance \(w\) is from \(F_2\) to the point on the edge. Evaluate \(s\), \(w\), and the total distance (\(s+w\)).

    Remember, the total distance when connecting

    1. The first focus \(F_1\)
    2. Any point on the edge
    3. The second focus \(F_2\)

    equals \(2a\). Based on this information, find the location of the right vertex at (\(x\), 0) in the diagram below.

    plot of chunk unnamed-chunk-4



    Solution


  7. Question

    An ellipse is a set of points such that the total distance from (1) one focus to (2) the edge to (3) the other focus is constant for any point on the edge. The two foci (\(F_1\) and \(F_2\)) are a distance \(2c\) apart, and the total distance from \(F_1\) to the edge to \(F_2\) is a distance \(2a\).

    Let’s consider the example below with foci at \((-2.1,0)\) and \((2.1,0)\) and a covertex at \((0, 7.2)\):

    plot of chunk unnamed-chunk-2

    Evaluate \(2a\) and \(2c\).

    Now, let’s pick another point on the ellipse, with (approximate) coordinates (4.3, 5.9):

    plot of chunk unnamed-chunk-3

    The distance \(s\) is from \(F_1\) to the point on the edge. The distance \(w\) is from \(F_2\) to the point on the edge. Evaluate \(s\), \(w\), and the total distance (\(s+w\)).

    Remember, the total distance when connecting

    1. The first focus \(F_1\)
    2. Any point on the edge
    3. The second focus \(F_2\)

    equals \(2a\). Based on this information, find the location of the right vertex at (\(x\), 0) in the diagram below.

    plot of chunk unnamed-chunk-4



    Solution


  8. Question

    An ellipse is a set of points such that the total distance from (1) one focus to (2) the edge to (3) the other focus is constant for any point on the edge. The two foci (\(F_1\) and \(F_2\)) are a distance \(2c\) apart, and the total distance from \(F_1\) to the edge to \(F_2\) is a distance \(2a\).

    Let’s consider the example below with foci at \((-4.5,0)\) and \((4.5,0)\) and a covertex at \((0, 2.8)\):

    plot of chunk unnamed-chunk-2

    Evaluate \(2a\) and \(2c\).

    Now, let’s pick another point on the ellipse, with (approximate) coordinates (4.59, 1.4):

    plot of chunk unnamed-chunk-3

    The distance \(s\) is from \(F_1\) to the point on the edge. The distance \(w\) is from \(F_2\) to the point on the edge. Evaluate \(s\), \(w\), and the total distance (\(s+w\)).

    Remember, the total distance when connecting

    1. The first focus \(F_1\)
    2. Any point on the edge
    3. The second focus \(F_2\)

    equals \(2a\). Based on this information, find the location of the right vertex at (\(x\), 0) in the diagram below.

    plot of chunk unnamed-chunk-4



    Solution


  9. Question

    An ellipse is a set of points such that the total distance from (1) one focus to (2) the edge to (3) the other focus is constant for any point on the edge. The two foci (\(F_1\) and \(F_2\)) are a distance \(2c\) apart, and the total distance from \(F_1\) to the edge to \(F_2\) is a distance \(2a\).

    Let’s consider the example below with foci at \((-6.5,0)\) and \((6.5,0)\) and a covertex at \((0, 7.2)\):

    plot of chunk unnamed-chunk-2

    Evaluate \(2a\) and \(2c\).

    Now, let’s pick another point on the ellipse, with (approximate) coordinates (8.96, 2.76):

    plot of chunk unnamed-chunk-3

    The distance \(s\) is from \(F_1\) to the point on the edge. The distance \(w\) is from \(F_2\) to the point on the edge. Evaluate \(s\), \(w\), and the total distance (\(s+w\)).

    Remember, the total distance when connecting

    1. The first focus \(F_1\)
    2. Any point on the edge
    3. The second focus \(F_2\)

    equals \(2a\). Based on this information, find the location of the right vertex at (\(x\), 0) in the diagram below.

    plot of chunk unnamed-chunk-4



    Solution


  10. Question

    An ellipse is a set of points such that the total distance from (1) one focus to (2) the edge to (3) the other focus is constant for any point on the edge. The two foci (\(F_1\) and \(F_2\)) are a distance \(2c\) apart, and the total distance from \(F_1\) to the edge to \(F_2\) is a distance \(2a\).

    Let’s consider the example below with foci at \((-2.8,0)\) and \((2.8,0)\) and a covertex at \((0, 9.6)\):

    plot of chunk unnamed-chunk-2

    Evaluate \(2a\) and \(2c\).

    Now, let’s pick another point on the ellipse, with (approximate) coordinates (4.99, 8.32):

    plot of chunk unnamed-chunk-3

    The distance \(s\) is from \(F_1\) to the point on the edge. The distance \(w\) is from \(F_2\) to the point on the edge. Evaluate \(s\), \(w\), and the total distance (\(s+w\)).

    Remember, the total distance when connecting

    1. The first focus \(F_1\)
    2. Any point on the edge
    3. The second focus \(F_2\)

    equals \(2a\). Based on this information, find the location of the right vertex at (\(x\), 0) in the diagram below.

    plot of chunk unnamed-chunk-4



    Solution


  11. Question

    An ellipse is the set of (\(x\), \(y\)) points that satisfy the equation below.

    \[\frac{(x-h)^2}{(r_1)^2}+\frac{(y-k)^2}{(r_2)^2}=1\]

    where the center is \((h,k)\), the horizontal radius is \(r_1\), and the vertical radius is \(r_2\).

    Determine the parameters of the ellipse in the graph below. You can assume all parameters are integers.

    plot of chunk unnamed-chunk-2

    Which of the following equations would give the graph above?

    \[\begin{align} A&:~~\frac{(x+5)^2}{4}+\frac{(y+6)^2}{9}=1 & B&:~~\frac{(x+5)^2}{4}+\frac{(y-6)^2}{9}=1 & C&:~~\frac{(x-5)^2}{4}+\frac{(y+6)^2}{9}=1 & D&:~~\frac{(x-5)^2}{4}+\frac{(y-6)^2}{9}=1 \\ E&:~~\frac{(x+6)^2}{4}+\frac{(y+5)^2}{9}=1 & F&:~~\frac{(x+6)^2}{4}+\frac{(y-5)^2}{9}=1 & G&:~~\frac{(x-6)^2}{4}+\frac{(y+5)^2}{9}=1 & H&:~~\frac{(x-6)^2}{4}+\frac{(y-5)^2}{9}=1 \\ I&:~~\frac{(x+5)^2}{9}+\frac{(y+6)^2}{4}=1 & J&:~~\frac{(x+5)^2}{9}+\frac{(y-6)^2}{4}=1 & K&:~~\frac{(x-5)^2}{9}+\frac{(y+6)^2}{4}=1 & L&:~~\frac{(x-5)^2}{9}+\frac{(y-6)^2}{4}=1 \\ M&:~~\frac{(x+6)^2}{9}+\frac{(y+5)^2}{4}=1 & N&:~~\frac{(x+6)^2}{9}+\frac{(y-5)^2}{4}=1 & O&:~~\frac{(x-6)^2}{9}+\frac{(y+5)^2}{4}=1 & P&:~~\frac{(x-6)^2}{9}+\frac{(y-5)^2}{4}=1 \\ \end{align}\]

    Equation



    Solution


  12. Question

    An ellipse is the set of (\(x\), \(y\)) points that satisfy the equation below.

    \[\frac{(x-h)^2}{(r_1)^2}+\frac{(y-k)^2}{(r_2)^2}=1\]

    where the center is \((h,k)\), the horizontal radius is \(r_1\), and the vertical radius is \(r_2\).

    Determine the parameters of the ellipse in the graph below. You can assume all parameters are integers.

    plot of chunk unnamed-chunk-2

    Which of the following equations would give the graph above?

    \[\begin{align} A&:~~\frac{(x+3)^2}{4}+\frac{(y+8)^2}{16}=1 & B&:~~\frac{(x+3)^2}{4}+\frac{(y-8)^2}{16}=1 & C&:~~\frac{(x-3)^2}{4}+\frac{(y+8)^2}{16}=1 & D&:~~\frac{(x-3)^2}{4}+\frac{(y-8)^2}{16}=1 \\ E&:~~\frac{(x+8)^2}{4}+\frac{(y+3)^2}{16}=1 & F&:~~\frac{(x+8)^2}{4}+\frac{(y-3)^2}{16}=1 & G&:~~\frac{(x-8)^2}{4}+\frac{(y+3)^2}{16}=1 & H&:~~\frac{(x-8)^2}{4}+\frac{(y-3)^2}{16}=1 \\ I&:~~\frac{(x+3)^2}{16}+\frac{(y+8)^2}{4}=1 & J&:~~\frac{(x+3)^2}{16}+\frac{(y-8)^2}{4}=1 & K&:~~\frac{(x-3)^2}{16}+\frac{(y+8)^2}{4}=1 & L&:~~\frac{(x-3)^2}{16}+\frac{(y-8)^2}{4}=1 \\ M&:~~\frac{(x+8)^2}{16}+\frac{(y+3)^2}{4}=1 & N&:~~\frac{(x+8)^2}{16}+\frac{(y-3)^2}{4}=1 & O&:~~\frac{(x-8)^2}{16}+\frac{(y+3)^2}{4}=1 & P&:~~\frac{(x-8)^2}{16}+\frac{(y-3)^2}{4}=1 \\ \end{align}\]

    Equation



    Solution


  13. Question

    An ellipse is the set of (\(x\), \(y\)) points that satisfy the equation below.

    \[\frac{(x-h)^2}{(r_1)^2}+\frac{(y-k)^2}{(r_2)^2}=1\]

    where the center is \((h,k)\), the horizontal radius is \(r_1\), and the vertical radius is \(r_2\).

    Determine the parameters of the ellipse in the graph below. You can assume all parameters are integers.

    plot of chunk unnamed-chunk-2

    Which of the following equations would give the graph above?

    \[\begin{align} A&:~~\frac{(x+7)^2}{4}+\frac{(y+8)^2}{9}=1 & B&:~~\frac{(x+7)^2}{4}+\frac{(y-8)^2}{9}=1 & C&:~~\frac{(x-7)^2}{4}+\frac{(y+8)^2}{9}=1 & D&:~~\frac{(x-7)^2}{4}+\frac{(y-8)^2}{9}=1 \\ E&:~~\frac{(x+8)^2}{4}+\frac{(y+7)^2}{9}=1 & F&:~~\frac{(x+8)^2}{4}+\frac{(y-7)^2}{9}=1 & G&:~~\frac{(x-8)^2}{4}+\frac{(y+7)^2}{9}=1 & H&:~~\frac{(x-8)^2}{4}+\frac{(y-7)^2}{9}=1 \\ I&:~~\frac{(x+7)^2}{9}+\frac{(y+8)^2}{4}=1 & J&:~~\frac{(x+7)^2}{9}+\frac{(y-8)^2}{4}=1 & K&:~~\frac{(x-7)^2}{9}+\frac{(y+8)^2}{4}=1 & L&:~~\frac{(x-7)^2}{9}+\frac{(y-8)^2}{4}=1 \\ M&:~~\frac{(x+8)^2}{9}+\frac{(y+7)^2}{4}=1 & N&:~~\frac{(x+8)^2}{9}+\frac{(y-7)^2}{4}=1 & O&:~~\frac{(x-8)^2}{9}+\frac{(y+7)^2}{4}=1 & P&:~~\frac{(x-8)^2}{9}+\frac{(y-7)^2}{4}=1 \\ \end{align}\]

    Equation



    Solution


  14. Question

    An ellipse is the set of (\(x\), \(y\)) points that satisfy the equation below.

    \[\frac{(x-h)^2}{(r_1)^2}+\frac{(y-k)^2}{(r_2)^2}=1\]

    where the center is \((h,k)\), the horizontal radius is \(r_1\), and the vertical radius is \(r_2\).

    Determine the parameters of the ellipse in the graph below. You can assume all parameters are integers.

    plot of chunk unnamed-chunk-2

    Which of the following equations would give the graph above?

    \[\begin{align} A&:~~\frac{(x+6)^2}{9}+\frac{(y+7)^2}{16}=1 & B&:~~\frac{(x+6)^2}{9}+\frac{(y-7)^2}{16}=1 & C&:~~\frac{(x-6)^2}{9}+\frac{(y+7)^2}{16}=1 & D&:~~\frac{(x-6)^2}{9}+\frac{(y-7)^2}{16}=1 \\ E&:~~\frac{(x+7)^2}{9}+\frac{(y+6)^2}{16}=1 & F&:~~\frac{(x+7)^2}{9}+\frac{(y-6)^2}{16}=1 & G&:~~\frac{(x-7)^2}{9}+\frac{(y+6)^2}{16}=1 & H&:~~\frac{(x-7)^2}{9}+\frac{(y-6)^2}{16}=1 \\ I&:~~\frac{(x+6)^2}{16}+\frac{(y+7)^2}{9}=1 & J&:~~\frac{(x+6)^2}{16}+\frac{(y-7)^2}{9}=1 & K&:~~\frac{(x-6)^2}{16}+\frac{(y+7)^2}{9}=1 & L&:~~\frac{(x-6)^2}{16}+\frac{(y-7)^2}{9}=1 \\ M&:~~\frac{(x+7)^2}{16}+\frac{(y+6)^2}{9}=1 & N&:~~\frac{(x+7)^2}{16}+\frac{(y-6)^2}{9}=1 & O&:~~\frac{(x-7)^2}{16}+\frac{(y+6)^2}{9}=1 & P&:~~\frac{(x-7)^2}{16}+\frac{(y-6)^2}{9}=1 \\ \end{align}\]

    Equation



    Solution


  15. Question

    An ellipse is the set of (\(x\), \(y\)) points that satisfy the equation below.

    \[\frac{(x-h)^2}{(r_1)^2}+\frac{(y-k)^2}{(r_2)^2}=1\]

    where the center is \((h,k)\), the horizontal radius is \(r_1\), and the vertical radius is \(r_2\).

    Determine the parameters of the ellipse in the graph below. You can assume all parameters are integers.

    plot of chunk unnamed-chunk-2

    Which of the following equations would give the graph above?

    \[\begin{align} A&:~~\frac{(x+5)^2}{4}+\frac{(y+6)^2}{9}=1 & B&:~~\frac{(x+5)^2}{4}+\frac{(y-6)^2}{9}=1 & C&:~~\frac{(x-5)^2}{4}+\frac{(y+6)^2}{9}=1 & D&:~~\frac{(x-5)^2}{4}+\frac{(y-6)^2}{9}=1 \\ E&:~~\frac{(x+6)^2}{4}+\frac{(y+5)^2}{9}=1 & F&:~~\frac{(x+6)^2}{4}+\frac{(y-5)^2}{9}=1 & G&:~~\frac{(x-6)^2}{4}+\frac{(y+5)^2}{9}=1 & H&:~~\frac{(x-6)^2}{4}+\frac{(y-5)^2}{9}=1 \\ I&:~~\frac{(x+5)^2}{9}+\frac{(y+6)^2}{4}=1 & J&:~~\frac{(x+5)^2}{9}+\frac{(y-6)^2}{4}=1 & K&:~~\frac{(x-5)^2}{9}+\frac{(y+6)^2}{4}=1 & L&:~~\frac{(x-5)^2}{9}+\frac{(y-6)^2}{4}=1 \\ M&:~~\frac{(x+6)^2}{9}+\frac{(y+5)^2}{4}=1 & N&:~~\frac{(x+6)^2}{9}+\frac{(y-5)^2}{4}=1 & O&:~~\frac{(x-6)^2}{9}+\frac{(y+5)^2}{4}=1 & P&:~~\frac{(x-6)^2}{9}+\frac{(y-5)^2}{4}=1 \\ \end{align}\]

    Equation



    Solution


  16. Question

    An ellipse is the set of (\(x\), \(y\)) points that satisfy the equation below.

    \[\frac{(x-h)^2}{(r_1)^2}+\frac{(y-k)^2}{(r_2)^2}=1\]

    where the center is \((h,k)\), the horizontal radius is \(r_1\), and the vertical radius is \(r_2\).

    Determine the parameters of the ellipse in the graph below. You can assume all parameters are integers.

    plot of chunk unnamed-chunk-2

    Which of the following equations would give the graph above?

    \[\begin{align} A&:~~\frac{(x+2)^2}{9}+\frac{(y+4)^2}{25}=1 & B&:~~\frac{(x+2)^2}{9}+\frac{(y-4)^2}{25}=1 & C&:~~\frac{(x-2)^2}{9}+\frac{(y+4)^2}{25}=1 & D&:~~\frac{(x-2)^2}{9}+\frac{(y-4)^2}{25}=1 \\ E&:~~\frac{(x+4)^2}{9}+\frac{(y+2)^2}{25}=1 & F&:~~\frac{(x+4)^2}{9}+\frac{(y-2)^2}{25}=1 & G&:~~\frac{(x-4)^2}{9}+\frac{(y+2)^2}{25}=1 & H&:~~\frac{(x-4)^2}{9}+\frac{(y-2)^2}{25}=1 \\ I&:~~\frac{(x+2)^2}{25}+\frac{(y+4)^2}{9}=1 & J&:~~\frac{(x+2)^2}{25}+\frac{(y-4)^2}{9}=1 & K&:~~\frac{(x-2)^2}{25}+\frac{(y+4)^2}{9}=1 & L&:~~\frac{(x-2)^2}{25}+\frac{(y-4)^2}{9}=1 \\ M&:~~\frac{(x+4)^2}{25}+\frac{(y+2)^2}{9}=1 & N&:~~\frac{(x+4)^2}{25}+\frac{(y-2)^2}{9}=1 & O&:~~\frac{(x-4)^2}{25}+\frac{(y+2)^2}{9}=1 & P&:~~\frac{(x-4)^2}{25}+\frac{(y-2)^2}{9}=1 \\ \end{align}\]

    Equation



    Solution


  17. Question

    An ellipse is the set of (\(x\), \(y\)) points that satisfy the equation below.

    \[\frac{(x-h)^2}{(r_1)^2}+\frac{(y-k)^2}{(r_2)^2}=1\]

    where the center is \((h,k)\), the horizontal radius is \(r_1\), and the vertical radius is \(r_2\).

    Determine the parameters of the ellipse in the graph below. You can assume all parameters are integers.

    plot of chunk unnamed-chunk-2

    Which of the following equations would give the graph above?

    \[\begin{align} A&:~~\frac{(x+5)^2}{9}+\frac{(y+6)^2}{16}=1 & B&:~~\frac{(x+5)^2}{9}+\frac{(y-6)^2}{16}=1 & C&:~~\frac{(x-5)^2}{9}+\frac{(y+6)^2}{16}=1 & D&:~~\frac{(x-5)^2}{9}+\frac{(y-6)^2}{16}=1 \\ E&:~~\frac{(x+6)^2}{9}+\frac{(y+5)^2}{16}=1 & F&:~~\frac{(x+6)^2}{9}+\frac{(y-5)^2}{16}=1 & G&:~~\frac{(x-6)^2}{9}+\frac{(y+5)^2}{16}=1 & H&:~~\frac{(x-6)^2}{9}+\frac{(y-5)^2}{16}=1 \\ I&:~~\frac{(x+5)^2}{16}+\frac{(y+6)^2}{9}=1 & J&:~~\frac{(x+5)^2}{16}+\frac{(y-6)^2}{9}=1 & K&:~~\frac{(x-5)^2}{16}+\frac{(y+6)^2}{9}=1 & L&:~~\frac{(x-5)^2}{16}+\frac{(y-6)^2}{9}=1 \\ M&:~~\frac{(x+6)^2}{16}+\frac{(y+5)^2}{9}=1 & N&:~~\frac{(x+6)^2}{16}+\frac{(y-5)^2}{9}=1 & O&:~~\frac{(x-6)^2}{16}+\frac{(y+5)^2}{9}=1 & P&:~~\frac{(x-6)^2}{16}+\frac{(y-5)^2}{9}=1 \\ \end{align}\]

    Equation



    Solution


  18. Question

    An ellipse is the set of (\(x\), \(y\)) points that satisfy the equation below.

    \[\frac{(x-h)^2}{(r_1)^2}+\frac{(y-k)^2}{(r_2)^2}=1\]

    where the center is \((h,k)\), the horizontal radius is \(r_1\), and the vertical radius is \(r_2\).

    Determine the parameters of the ellipse in the graph below. You can assume all parameters are integers.

    plot of chunk unnamed-chunk-2

    Which of the following equations would give the graph above?

    \[\begin{align} A&:~~\frac{(x+2)^2}{16}+\frac{(y+5)^2}{64}=1 & B&:~~\frac{(x+2)^2}{16}+\frac{(y-5)^2}{64}=1 & C&:~~\frac{(x-2)^2}{16}+\frac{(y+5)^2}{64}=1 & D&:~~\frac{(x-2)^2}{16}+\frac{(y-5)^2}{64}=1 \\ E&:~~\frac{(x+5)^2}{16}+\frac{(y+2)^2}{64}=1 & F&:~~\frac{(x+5)^2}{16}+\frac{(y-2)^2}{64}=1 & G&:~~\frac{(x-5)^2}{16}+\frac{(y+2)^2}{64}=1 & H&:~~\frac{(x-5)^2}{16}+\frac{(y-2)^2}{64}=1 \\ I&:~~\frac{(x+2)^2}{64}+\frac{(y+5)^2}{16}=1 & J&:~~\frac{(x+2)^2}{64}+\frac{(y-5)^2}{16}=1 & K&:~~\frac{(x-2)^2}{64}+\frac{(y+5)^2}{16}=1 & L&:~~\frac{(x-2)^2}{64}+\frac{(y-5)^2}{16}=1 \\ M&:~~\frac{(x+5)^2}{64}+\frac{(y+2)^2}{16}=1 & N&:~~\frac{(x+5)^2}{64}+\frac{(y-2)^2}{16}=1 & O&:~~\frac{(x-5)^2}{64}+\frac{(y+2)^2}{16}=1 & P&:~~\frac{(x-5)^2}{64}+\frac{(y-2)^2}{16}=1 \\ \end{align}\]

    Equation



    Solution


  19. Question

    An ellipse is the set of (\(x\), \(y\)) points that satisfy the equation below.

    \[\frac{(x-h)^2}{(r_1)^2}+\frac{(y-k)^2}{(r_2)^2}=1\]

    where the center is \((h,k)\), the horizontal radius is \(r_1\), and the vertical radius is \(r_2\).

    Determine the parameters of the ellipse in the graph below. You can assume all parameters are integers.

    plot of chunk unnamed-chunk-2

    Which of the following equations would give the graph above?

    \[\begin{align} A&:~~\frac{(x+4)^2}{4}+\frac{(y+8)^2}{9}=1 & B&:~~\frac{(x+4)^2}{4}+\frac{(y-8)^2}{9}=1 & C&:~~\frac{(x-4)^2}{4}+\frac{(y+8)^2}{9}=1 & D&:~~\frac{(x-4)^2}{4}+\frac{(y-8)^2}{9}=1 \\ E&:~~\frac{(x+8)^2}{4}+\frac{(y+4)^2}{9}=1 & F&:~~\frac{(x+8)^2}{4}+\frac{(y-4)^2}{9}=1 & G&:~~\frac{(x-8)^2}{4}+\frac{(y+4)^2}{9}=1 & H&:~~\frac{(x-8)^2}{4}+\frac{(y-4)^2}{9}=1 \\ I&:~~\frac{(x+4)^2}{9}+\frac{(y+8)^2}{4}=1 & J&:~~\frac{(x+4)^2}{9}+\frac{(y-8)^2}{4}=1 & K&:~~\frac{(x-4)^2}{9}+\frac{(y+8)^2}{4}=1 & L&:~~\frac{(x-4)^2}{9}+\frac{(y-8)^2}{4}=1 \\ M&:~~\frac{(x+8)^2}{9}+\frac{(y+4)^2}{4}=1 & N&:~~\frac{(x+8)^2}{9}+\frac{(y-4)^2}{4}=1 & O&:~~\frac{(x-8)^2}{9}+\frac{(y+4)^2}{4}=1 & P&:~~\frac{(x-8)^2}{9}+\frac{(y-4)^2}{4}=1 \\ \end{align}\]

    Equation



    Solution


  20. Question

    An ellipse is the set of (\(x\), \(y\)) points that satisfy the equation below.

    \[\frac{(x-h)^2}{(r_1)^2}+\frac{(y-k)^2}{(r_2)^2}=1\]

    where the center is \((h,k)\), the horizontal radius is \(r_1\), and the vertical radius is \(r_2\).

    Determine the parameters of the ellipse in the graph below. You can assume all parameters are integers.

    plot of chunk unnamed-chunk-2

    Which of the following equations would give the graph above?

    \[\begin{align} A&:~~\frac{(x+2)^2}{9}+\frac{(y+5)^2}{49}=1 & B&:~~\frac{(x+2)^2}{9}+\frac{(y-5)^2}{49}=1 & C&:~~\frac{(x-2)^2}{9}+\frac{(y+5)^2}{49}=1 & D&:~~\frac{(x-2)^2}{9}+\frac{(y-5)^2}{49}=1 \\ E&:~~\frac{(x+5)^2}{9}+\frac{(y+2)^2}{49}=1 & F&:~~\frac{(x+5)^2}{9}+\frac{(y-2)^2}{49}=1 & G&:~~\frac{(x-5)^2}{9}+\frac{(y+2)^2}{49}=1 & H&:~~\frac{(x-5)^2}{9}+\frac{(y-2)^2}{49}=1 \\ I&:~~\frac{(x+2)^2}{49}+\frac{(y+5)^2}{9}=1 & J&:~~\frac{(x+2)^2}{49}+\frac{(y-5)^2}{9}=1 & K&:~~\frac{(x-2)^2}{49}+\frac{(y+5)^2}{9}=1 & L&:~~\frac{(x-2)^2}{49}+\frac{(y-5)^2}{9}=1 \\ M&:~~\frac{(x+5)^2}{49}+\frac{(y+2)^2}{9}=1 & N&:~~\frac{(x+5)^2}{49}+\frac{(y-2)^2}{9}=1 & O&:~~\frac{(x-5)^2}{49}+\frac{(y+2)^2}{9}=1 & P&:~~\frac{(x-5)^2}{49}+\frac{(y-2)^2}{9}=1 \\ \end{align}\]

    Equation



    Solution


  21. Question

    A gardener is making an ellipse using the pins and string method. She places the two stakes (pins) 16 meters apart, and she uses a string that is 20 meters long (ignoring the amount needed to tie around the stake).

    How long is the major axis of the generated ellipse?


    Solution


  22. Question

    A gardener is making an ellipse using the pins and string method. She places the two stakes (pins) 4.8 meters apart, and she uses a string that is 14.8 meters long (ignoring the amount needed to tie around the stake).

    How long is the major axis of the generated ellipse?


    Solution


  23. Question

    A gardener is making an ellipse using the pins and string method. She places the two stakes (pins) 11.2 meters apart, and she uses a string that is 13 meters long (ignoring the amount needed to tie around the stake).

    How long is the major axis of the generated ellipse?


    Solution


  24. Question

    A gardener is making an ellipse using the pins and string method. She places the two stakes (pins) 3.2 meters apart, and she uses a string that is 13 meters long (ignoring the amount needed to tie around the stake).

    How long is the major axis of the generated ellipse?


    Solution


  25. Question

    A gardener is making an ellipse using the pins and string method. She places the two stakes (pins) 2.6 meters apart, and she uses a string that is 17 meters long (ignoring the amount needed to tie around the stake).

    How long is the major axis of the generated ellipse?


    Solution


  26. Question

    A gardener is making an ellipse using the pins and string method. She places the two stakes (pins) 14.4 meters apart, and she uses a string that is 18 meters long (ignoring the amount needed to tie around the stake).

    How long is the major axis of the generated ellipse?


    Solution


  27. Question

    A gardener is making an ellipse using the pins and string method. She places the two stakes (pins) 2.6 meters apart, and she uses a string that is 17 meters long (ignoring the amount needed to tie around the stake).

    How long is the major axis of the generated ellipse?


    Solution


  28. Question

    A gardener is making an ellipse using the pins and string method. She places the two stakes (pins) 13 meters apart, and she uses a string that is 19.4 meters long (ignoring the amount needed to tie around the stake).

    How long is the major axis of the generated ellipse?


    Solution


  29. Question

    A gardener is making an ellipse using the pins and string method. She places the two stakes (pins) 6.6 meters apart, and she uses a string that is 13 meters long (ignoring the amount needed to tie around the stake).

    How long is the major axis of the generated ellipse?


    Solution


  30. Question

    A gardener is making an ellipse using the pins and string method. She places the two stakes (pins) 11 meters apart, and she uses a string that is 14.6 meters long (ignoring the amount needed to tie around the stake).

    How long is the major axis of the generated ellipse?


    Solution


  31. Question

    A gardener is making an ellipse using the pins and string method. She places the two stakes (pins) 9.6 meters apart, and she uses a string that is 14.6 meters long (ignoring the amount needed to tie around the stake).

    How long is the minor axis of the generated ellipse?


    Solution


  32. Question

    A gardener is making an ellipse using the pins and string method. She places the two stakes (pins) 3.2 meters apart, and she uses a string that is 13 meters long (ignoring the amount needed to tie around the stake).

    How long is the minor axis of the generated ellipse?


    Solution


  33. Question

    A gardener is making an ellipse using the pins and string method. She places the two stakes (pins) 2.2 meters apart, and she uses a string that is 12.2 meters long (ignoring the amount needed to tie around the stake).

    How long is the minor axis of the generated ellipse?


    Solution


  34. Question

    A gardener is making an ellipse using the pins and string method. She places the two stakes (pins) 13 meters apart, and she uses a string that is 19.4 meters long (ignoring the amount needed to tie around the stake).

    How long is the minor axis of the generated ellipse?


    Solution


  35. Question

    A gardener is making an ellipse using the pins and string method. She places the two stakes (pins) 10.8 meters apart, and she uses a string that is 18 meters long (ignoring the amount needed to tie around the stake).

    How long is the minor axis of the generated ellipse?


    Solution


  36. Question

    A gardener is making an ellipse using the pins and string method. She places the two stakes (pins) 12.8 meters apart, and she uses a string that is 16 meters long (ignoring the amount needed to tie around the stake).

    How long is the minor axis of the generated ellipse?


    Solution


  37. Question

    A gardener is making an ellipse using the pins and string method. She places the two stakes (pins) 5.6 meters apart, and she uses a string that is 10.6 meters long (ignoring the amount needed to tie around the stake).

    How long is the minor axis of the generated ellipse?


    Solution


  38. Question

    A gardener is making an ellipse using the pins and string method. She places the two stakes (pins) 7.8 meters apart, and she uses a string that is 17.8 meters long (ignoring the amount needed to tie around the stake).

    How long is the minor axis of the generated ellipse?


    Solution


  39. Question

    A gardener is making an ellipse using the pins and string method. She places the two stakes (pins) 9.6 meters apart, and she uses a string that is 14.6 meters long (ignoring the amount needed to tie around the stake).

    How long is the minor axis of the generated ellipse?


    Solution


  40. Question

    A gardener is making an ellipse using the pins and string method. She places the two stakes (pins) 12.8 meters apart, and she uses a string that is 16 meters long (ignoring the amount needed to tie around the stake).

    How long is the minor axis of the generated ellipse?


    Solution


  41. Question

    A gardener is making an ellipse using the pins and string method. She wants the length of the major axis to be 17 meters and the length of the minor axis to be 16.8 meters.

    How long should the string be?


    Solution


  42. Question

    A gardener is making an ellipse using the pins and string method. She wants the length of the major axis to be 10.6 meters and the length of the minor axis to be 9 meters.

    How long should the string be?


    Solution


  43. Question

    A gardener is making an ellipse using the pins and string method. She wants the length of the major axis to be 14.6 meters and the length of the minor axis to be 9.6 meters.

    How long should the string be?


    Solution


  44. Question

    A gardener is making an ellipse using the pins and string method. She wants the length of the major axis to be 19.4 meters and the length of the minor axis to be 14.4 meters.

    How long should the string be?


    Solution


  45. Question

    A gardener is making an ellipse using the pins and string method. She wants the length of the major axis to be 10.6 meters and the length of the minor axis to be 5.6 meters.

    How long should the string be?


    Solution


  46. Question

    A gardener is making an ellipse using the pins and string method. She wants the length of the major axis to be 17.8 meters and the length of the minor axis to be 16 meters.

    How long should the string be?


    Solution


  47. Question

    A gardener is making an ellipse using the pins and string method. She wants the length of the major axis to be 17 meters and the length of the minor axis to be 15.4 meters.

    How long should the string be?


    Solution


  48. Question

    A gardener is making an ellipse using the pins and string method. She wants the length of the major axis to be 17 meters and the length of the minor axis to be 15.4 meters.

    How long should the string be?


    Solution


  49. Question

    A gardener is making an ellipse using the pins and string method. She wants the length of the major axis to be 13 meters and the length of the minor axis to be 12.6 meters.

    How long should the string be?


    Solution


  50. Question

    A gardener is making an ellipse using the pins and string method. She wants the length of the major axis to be 17 meters and the length of the minor axis to be 15 meters.

    How long should the string be?


    Solution


  51. Question

    A gardener is making an ellipse using the pins and string method. She wants the length of the major axis to be 17.8 meters and the length of the minor axis to be 7.8 meters.

    How far apart should the stakes be placed?


    Solution


  52. Question

    A gardener is making an ellipse using the pins and string method. She wants the length of the major axis to be 19.4 meters and the length of the minor axis to be 13 meters.

    How far apart should the stakes be placed?


    Solution


  53. Question

    A gardener is making an ellipse using the pins and string method. She wants the length of the major axis to be 10.6 meters and the length of the minor axis to be 9 meters.

    How far apart should the stakes be placed?


    Solution


  54. Question

    A gardener is making an ellipse using the pins and string method. She wants the length of the major axis to be 13 meters and the length of the minor axis to be 11.2 meters.

    How far apart should the stakes be placed?


    Solution


  55. Question

    A gardener is making an ellipse using the pins and string method. She wants the length of the major axis to be 10.6 meters and the length of the minor axis to be 5.6 meters.

    How far apart should the stakes be placed?


    Solution


  56. Question

    A gardener is making an ellipse using the pins and string method. She wants the length of the major axis to be 13 meters and the length of the minor axis to be 6.6 meters.

    How far apart should the stakes be placed?


    Solution


  57. Question

    A gardener is making an ellipse using the pins and string method. She wants the length of the major axis to be 13 meters and the length of the minor axis to be 12.6 meters.

    How far apart should the stakes be placed?


    Solution


  58. Question

    A gardener is making an ellipse using the pins and string method. She wants the length of the major axis to be 17.8 meters and the length of the minor axis to be 16 meters.

    How far apart should the stakes be placed?


    Solution


  59. Question

    A gardener is making an ellipse using the pins and string method. She wants the length of the major axis to be 17 meters and the length of the minor axis to be 16.8 meters.

    How far apart should the stakes be placed?


    Solution


  60. Question

    A gardener is making an ellipse using the pins and string method. She wants the length of the major axis to be 19 meters and the length of the minor axis to be 15.2 meters.

    How far apart should the stakes be placed?


    Solution


  61. Question

    The following equation (in polynomial form) represents an ellipse.

    \[36x^2+72x+4y^2-24y-72=0\]

    With some algebra (completing the square), you can convert the equation into standard form:

    \[\frac{(x-h)^2}{(r_1)^2}+\frac{(y-k)^2}{(r_2)^2}=1\]

    From the given polynomial form, determine the 4 parameters.



    Solution


  62. Question

    The following equation (in polynomial form) represents an ellipse.

    \[49x^2+196x+25y^2-50y-1004=0\]

    With some algebra (completing the square), you can convert the equation into standard form:

    \[\frac{(x-h)^2}{(r_1)^2}+\frac{(y-k)^2}{(r_2)^2}=1\]

    From the given polynomial form, determine the 4 parameters.



    Solution


  63. Question

    The following equation (in polynomial form) represents an ellipse.

    \[16x^2-32x+36y^2-144y-416=0\]

    With some algebra (completing the square), you can convert the equation into standard form:

    \[\frac{(x-h)^2}{(r_1)^2}+\frac{(y-k)^2}{(r_2)^2}=1\]

    From the given polynomial form, determine the 4 parameters.



    Solution


  64. Question

    The following equation (in polynomial form) represents an ellipse.

    \[25x^2-150x+36y^2+72y-639=0\]

    With some algebra (completing the square), you can convert the equation into standard form:

    \[\frac{(x-h)^2}{(r_1)^2}+\frac{(y-k)^2}{(r_2)^2}=1\]

    From the given polynomial form, determine the 4 parameters.



    Solution


  65. Question

    The following equation (in polynomial form) represents an ellipse.

    \[4x^2+24x+16y^2-256y+996=0\]

    With some algebra (completing the square), you can convert the equation into standard form:

    \[\frac{(x-h)^2}{(r_1)^2}+\frac{(y-k)^2}{(r_2)^2}=1\]

    From the given polynomial form, determine the 4 parameters.



    Solution


  66. Question

    The following equation (in polynomial form) represents an ellipse.

    \[9x^2+18x+81y^2-648y+576=0\]

    With some algebra (completing the square), you can convert the equation into standard form:

    \[\frac{(x-h)^2}{(r_1)^2}+\frac{(y-k)^2}{(r_2)^2}=1\]

    From the given polynomial form, determine the 4 parameters.



    Solution


  67. Question

    The following equation (in polynomial form) represents an ellipse.

    \[9x^2+18x+36y^2+504y+1449=0\]

    With some algebra (completing the square), you can convert the equation into standard form:

    \[\frac{(x-h)^2}{(r_1)^2}+\frac{(y-k)^2}{(r_2)^2}=1\]

    From the given polynomial form, determine the 4 parameters.



    Solution


  68. Question

    The following equation (in polynomial form) represents an ellipse.

    \[16x^2-32x+36y^2-360y+340=0\]

    With some algebra (completing the square), you can convert the equation into standard form:

    \[\frac{(x-h)^2}{(r_1)^2}+\frac{(y-k)^2}{(r_2)^2}=1\]

    From the given polynomial form, determine the 4 parameters.



    Solution


  69. Question

    The following equation (in polynomial form) represents an ellipse.

    \[36x^2-288x+9y^2+18y+261=0\]

    With some algebra (completing the square), you can convert the equation into standard form:

    \[\frac{(x-h)^2}{(r_1)^2}+\frac{(y-k)^2}{(r_2)^2}=1\]

    From the given polynomial form, determine the 4 parameters.



    Solution


  70. Question

    The following equation (in polynomial form) represents an ellipse.

    \[9x^2+72x+36y^2+72y-144=0\]

    With some algebra (completing the square), you can convert the equation into standard form:

    \[\frac{(x-h)^2}{(r_1)^2}+\frac{(y-k)^2}{(r_2)^2}=1\]

    From the given polynomial form, determine the 4 parameters.



    Solution